L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 49 L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 We hve looked t severl chrcteristics of the log function. Now, we will look t some of the properties of logs. Remember, since log is defined like this: y log x if nd only if x we cn sy tht log is n exponent. So, we would expect the properties of logs to be similr to the properties of exponents. y
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 50 r Find log. This is function, log, operting on its inverse, r. Recll tht 1 1 1 1 f f x f f x x f f x f f x x The function nd its inverse "undo" ech other. Thus, log r r log Find M. exponent This is function, operting on its inverse, log. The function nd inverse cncel. Thus, log M M Find log 1. We cn write 1 s 0 since cnnot be 0 (it is the bse of log). 0 Thus, log 1 log 0 Recll tht the grph of the log psses through the point (1, 0). Find log. We cn write s 1 1. Thus, log log 1 Find log MN Here, we need to be bit clever. Let s see wht hppens if we let A log M nd B log N. We cn convert ech of these into exponentils: A B A log M M nd B log N N So, we cn write log MN log A B log A B A B log M log N Thus, log MN log M log N In words, the log of product is the sum of the individul logs. A B AB Note the corresponding exponent property:. It looks like the log property in reverse.
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 51 M Find log N Let A log M nd B log N. We cn convert ech of these into exponentils: A B A log M M nd B log N N So, we cn write A M AB log log log A B log M log N B N M Thus, log log M log N N In words, the log of quotient is the difference of the individul logs. A AB Note the corresponding exponent property:. It looks like the log property in reverse. B Find log Let M r A log M A We cn convert this to n exponentil: A log M M r A ra log M log log ra r log M So, we cn write r In words, constnt times log cn be mde the exponent of the rgument.
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 5 Using these properties of logrithms enbled people to replce difficult opertions on numbers with esier opertions on their logrithms. People in the old dys constructed "log tbles" tht could be used to quickly look up the log or ntilog (power of 10) of number. These were printed in books. We tught students how to use these tbles until the 1970's when hnd held clcultors becme chep enough for students to buy.
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 53 One key ppliction of these techniques ws celestil nvigtion. Once the invention of the chronometer mde possible the ccurte mesurement of longitude t se, mriners hd everything necessry to reduce their nvigtionl computtions to mere dditions. A five-digit tble of logrithms nd tble of the logrithms of trigonometric functions sufficed for most purposes, nd those tbles could fit in smll book. A fst wy of doing such clcultions ws using slide rule. Tht ws simply two sticks with log scles printed on them tht one would slide bck nd forth to show ddition of logs to do multipliction or subtrction of logs to do division.
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 54 There re couple more properties of logs tht we will find very useful. Here is property we hve seen for exponentil equtions: Tht is, if two exponentils re equl nd the bses re equl, then the exponents re equl. 4 4 1 x 4 There is corresponding property for logs. We cn tke log of both sides of n eqution nd retin equlity (s long s domins re not violted). 16 ln ln 16 ln ln16 ln16 ln x 4 ln 16 ln ln 16 ln x 3.7757 x 0.6931471 3
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 55 Let y y log M. Then, we cn write M Tke the log of both sides using log of ny bse: log b y log M b log b b log M b y log y log M b log M
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 56 We hve n exponentil bse e but log bse e. Let s rewrite this: log 9 log 3 log 3 e e e e e e e log 3 e Now, we hve n exponentil bse e nd log bse e, which re inverses nd so cncel ech log 3 e other. So, e 3
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 57 log log 9 log 9 3 log 3 log 3 log 3 log log3 log3 log3 log3 OR, use the chnge of bse formul: ln ln9 log log 9 3 ln3 ln ln ln3 ln 3 ln ln3 ln 3
L35-Wed-3-Nov-016-Sec-5-5-Properties-of-Logs-HW36-Moodle-Q9 pge 5 log x log x 1 log x log x 1 log x 1 log x 1 4 log 4 log x log x 1 log x 1 log log x log x 1 log x 1 1 log 1 log log 1 x x x 1 x x x log 1 log log 1 x 1 log log x 1 1 x x 1 x 1 log x log x 3log x log x 3log x 4 log 4 log x 3log x 1 3log log x x 3 1 x x 3 1 x x log log log